The moduli space of two-convex embedded spheres

TL;DR

This paper proves that the space of all 2-convex embedded spheres in Euclidean space is connected through continuous transformations, using mean curvature flow with surgery, extending ideas from scalar curvature geometry.

Contribution

It establishes the path-connectedness of the moduli space of 2-convex embedded spheres in any dimension, employing mean curvature flow techniques.

Findings

The moduli space of 2-convex embedded spheres is path-connected for all dimensions.

Mean curvature flow with surgery is effective in studying the topology of embedding spaces.

Analogous to scalar curvature results, this work extends geometric analysis methods to embedded spheres.

Abstract

We prove that the moduli space of 2-convex embedded n-spheres in R^{n+1} is path-connected for every n. Our proof uses mean curvature flow with surgery and can be seen as an extrinsic analog to Marques' influential proof of the path-connectedness of the moduli space of positive scalar curvature metics on three-manifolds.